We extend the theory of spinor class fields and relative spinor class fields to study
representation problems in several classical linear algebraic groups over number fields.
We apply this theory to study the set of isomorphism classes of maximal orders of central
simple algebras containing a given maximal Abelian suborder. We also study isometric
embeddings of one skew-Hermitian Quaternionic lattice into another.

The purpose of this paper, which is a continuation of [2, 3], is to prove further results about arithmetic modular forms and functions. In particular we shall demonstrate here a q-expansion principle which will be useful in proving a reciprocity law for special values of arithmetic Hilbert modular functions, of which the classical results on complex multiplication are a special case. The main feature of our treatment is, perhaps, its independence of the theory of abelian varieties.